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Industrially relevant constrained optimization problems, such as portfolio optimization and portfolio rebalancing, are often intractable or difficult to solve exactly. In this work, we propose and benchmark a decomposition pipeline targeting portfolio optimization and rebalancing problems with constraints. The pipeline decomposes the optimization problem into constrained subproblems, which are then solved separately and aggregated to give a final result. Our pipeline includes three main components: preprocessing of correlation matrices based on random matrix theory, modified spectral clustering based on Newman's algorithm, and risk rebalancing. Our empirical results show that our pipeline consistently decomposes real-world portfolio optimization problems into subproblems with a size reduction of approximately 80%. Since subproblems are then solved independently, our pipeline drastically reduces the total computation time for state-of-the-art solvers. Moreover, by decomposing large problems into several smaller subproblems, the pipeline enables the use of near-term quantum devices as solvers, providing a path toward practical utility of quantum computers in portfolio optimization.

We consider asymmetric simple exclusion processes with $N$ particles on the one-dimensional discrete torus with $L$ sites with following properties: (i) nearest-neighbor jumps on the torus, (ii) the jump rates depend only on the distance to the next particle in the direction of the jump, (iii) the jump rates are independent of $N$ and $L$. For measures with a long-range two-body interaction potential that depends only on the distance between neighboring particles we prove a relation between the interaction potential and particle jump rates that is necessary and sufficient for the measure to be invariant for the process. The normalization of the measure and the stationary current are computed both for finite $L$ and $N$ and in the thermodynamic limit. For a finitely many particles that evolve on $\mathbb{Z}$ with totally asymmetric jumps it is proved, using reverse duality, that a certain family of nonstationary measures with a microscopic shock and antishock evolves into a convex combination of such measures with weights given by random walk transition probabilities. On macroscopic scale this domain random walk is a travelling wave phenomenon tantamount to phase separation with a stable shock and stable antishock. Various potential applications of this result and open questions are outlined., Comment: 47 pages, 2 figures

We examine three non-conservative interacting particle systems in one dimension modeling epidemic spreading: the diffusive contact process (DCP), a model that we introduce and call generalized diffusive contact process (GDCP), and the susceptible-infectious-recovered (SIR) model. For each system, we establish duality relations through an analytical approach. As an application, we provide a new expression of the correlation functions for the steady state of the finite volume DCP and GDCP in contact with boundary reservoirs. We explicitly compute the one point correlation function of the steady state of a small sized DCP, and for arbitrary size in a particular setting of the GDCP. For the infinite SIR model, the duality relation obtained yields an explicit expression of the time evolution of some specific correlation functions, describing the time decay of the sizes of chains of susceptible individuals.

The projected sensitivity of the effective electron neutrino-mass measurement with the KATRIN experiment is below 0.3 eV (90 % CL) after five years of data acquisition. The sensitivity is affected by the increased rate of the background electrons from KATRIN's main spectrometer. A special shifted-analysing-plane (SAP) configuration was developed to reduce this background by a factor of two. The complex layout of electromagnetic fields in the SAP configuration requires a robust method of estimating these fields. We present in this paper a dedicated calibration measurement of the fields using conversion electrons of gaseous $^\mathrm{83m}$Kr, which enables the neutrino-mass measurements in the SAP configuration., Comment: 19 pages, 11 figures

Medical doctors rely on images of the human anatomy, such as magnetic resonance imaging (MRI), to localize regions of interest in the patient during diagnosis and treatment. Despite advances in medical imaging technology, the information conveyance remains unimodal. This visual representation fails to capture the complexity of the real, multisensory interaction with human tissue. However, perceiving multimodal information about the patient's anatomy and disease in real-time is critical for the success of medical procedures and patient outcome. We introduce a Multimodal Medical Image Interaction (MMII) framework to allow medical experts a dynamic, audiovisual interaction with human tissue in three-dimensional space. In a virtual reality environment, the user receives physically informed audiovisual feedback to improve the spatial perception of anatomical structures. MMII uses a model-based sonification approach to generate sounds derived from the geometry and physical properties of tissue, thereby eliminating the need for hand-crafted sound design. Two user studies involving 34 general and nine clinical experts were conducted to evaluate the proposed interaction framework's learnability, usability, and accuracy. Our results showed excellent learnability of audiovisual correspondence as the rate of correct associations significantly improved (p < 0.001) over the course of the study. MMII resulted in superior brain tumor localization accuracy (p < 0.05) compared to conventional medical image interaction. Our findings substantiate the potential of this novel framework to enhance interaction with medical images, for example, during surgical procedures where immediate and precise feedback is needed., Comment: Accepted for publication in IEEE TVCG; presentation at IEEE ISMAR 2024

We consider a Markovian model for the kinetics of RNA Polymerase (RNAP) which provides a physical explanation for the phenomenon of cooperative pushing during transcription elongation observed in biochemical experiments on Escherichia coli and yeast RNAP. To study how backtracking of RNAP affects cooperative pushing we incorporate into this model backward (upstream) RNAP moves. With a rigorous mathematical treatment of the model we derive conditions on the mutual static and kinetic interactions between RNAP under which backtracking preserves cooperative pushing. This is achieved by exact computation of several key properties in the steady state of this model, including the distribution of headway between two RNAP along the DNA template and the average RNAP velocity and flux.

This study focuses on advancing metascintillators to break the 100 ps barrier and approach the 10 ps target. We exploit nanophotonic features, specifically the Purcell effect, to shape and enhance the scintillation properties of the first-generation metascintillator. We demonstrate that a faster emission is achievable along with a more efficient conversion efficiency. This results in a coincidence time resolution improved by a factor of 1.6, crucial for TOF-PET applications.

The fact that neutrinos carry a non-vanishing rest mass is evidence of physics beyond the Standard Model of elementary particles. Their absolute mass bears important relevance from particle physics to cosmology. In this work, we report on the search for the effective electron antineutrino mass with the KATRIN experiment. KATRIN performs precision spectroscopy of the tritium $\beta$-decay close to the kinematic endpoint. Based on the first five neutrino-mass measurement campaigns, we derive a best-fit value of $m_\nu^{2} = {-0.14^{+0.13}_{-0.15}}~\mathrm{eV^2}$, resulting in an upper limit of $m_\nu < {0.45}~\mathrm{eV}$ at 90 % confidence level. With six times the statistics of previous data sets, amounting to 36 million electrons collected in 259 measurement days, a substantial reduction of the background level and improved systematic uncertainties, this result tightens KATRIN's previous bound by a factor of almost two., Comment: 61 pages, 20 figures, 2 tables

A search for full energy depositions from bosonic keV-scale dark matter candidates of masses between 65 keV and 1021 keV has been performed with data collected during Phase II of the GERmanium Detector Array (GERDA) experiment. Our analysis includes direct dark matter absorption as well as dark Compton scattering. With a total exposure of 105.5 kg yr, no evidence for a signal above the background has been observed. The resulting exclusion limits deduced with either Bayesian or Frequentist statistics are the most stringent direct constraints in the major part of the 140-1021 keV mass range. As an example, at a mass of 150 keV the dimensionless coupling of dark photons and axion-like particles to electrons has been constrained to $\alpha$'/$\alpha$ < 8.7x10$^{-24}$ and g$_{ae}$ < 3.3x10$^{-12}$ at 90% credible interval (CI), respectively. Additionally, a search for peak-like signals from beyond the Standard Model decays of nucleons and electrons is performed. We find for the inclusive decay of a single neutron in $^{76}$Ge a lower lifetime limit of $\tau_n$ > 1.5x10$^{24}$ yr and for a proton $\tau_p$ > 1.3x10$^{24}$ yr at 90% CI. For the electron decay e$^-\rightarrow\nu_e\gamma$ a lower limit of $\tau_e$ > 5.4x10$^{25}$ yr at 90% CI has been determined., Comment: 20 pages, 12 figures, 7 tables

Despite substantial progress in machine learning for scientific discovery in recent years, truly de novo design of small molecules which exhibit a property of interest remains a significant challenge. We introduce LambdaZero, a generative active learning approach to search for synthesizable molecules. Powered by deep reinforcement learning, LambdaZero learns to search over the vast space of molecules to discover candidates with a desired property. We apply LambdaZero with molecular docking to design novel small molecules that inhibit the enzyme soluble Epoxide Hydrolase 2 (sEH), while enforcing constraints on synthesizability and drug-likeliness. LambdaZero provides an exponential speedup in terms of the number of calls to the expensive molecular docking oracle, and LambdaZero de novo designed molecules reach docking scores that would otherwise require the virtual screening of a hundred billion molecules. Importantly, LambdaZero discovers novel scaffolds of synthesizable, drug-like inhibitors for sEH. In in vitro experimental validation, a series of ligands from a generated quinazoline-based scaffold were synthesized, and the lead inhibitor N-(4,6-di(pyrrolidin-1-yl)quinazolin-2-yl)-N-methylbenzamide (UM0152893) displayed sub-micromolar enzyme inhibition of sEH.

DNA-encoded libraries (DELs) are a powerful approach for rapidly screening large numbers of diverse compounds. One of the key challenges in using DELs is library design, which involves choosing the building blocks that will be combinatorially combined to produce the final library. In this paper we consider the task of protein-protein interaction (PPI) biased DEL design. To this end, we evaluate several machine learning algorithms on the PPI modulation task and use them as a reward for the proposed GFlowNet-based generative approach. We additionally investigate the possibility of using structural information about building blocks to design a hierarchical action space for the GFlowNet. The observed results indicate that GFlowNets are a promising approach for generating diverse combinatorial library candidates.

Reference Evapotranspiration (ET0) is a key parameter for designing smart irrigation scheduling, since it is related by a coefficient to the water needs of a crop. The United Nations Food and Agriculture Organization, proposed a standard method for ET0 computation (FAO56PM), based on the parameterization of the Penman-Monteith equation, that is widely adopted in the literature. To compute ET0 using the FAO56-PM method, four main weather parameters are needed: temperature, humidity, wind, and solar radiation (SR). One way to make daily ET0 estimations for future days is to use freely available weather forecast services (WFSs), where many meteorological parameters are estimated up to the next 15 days. A problem with this method is that currently, SR is not provided as a free forecast parameter on most of those online services or, normally, such forecasts present a financial cost penalty. For this reason, several ET0 estimation models using machine and deep learning were developed and presented in the literature, that use as input features a reduced set of carefully selected weather parameters, that are compatible with common freely available WFSs. However, most studies on this topic have only evaluated model performance using data from weather stations (WSs), without considering the effect of using weather forecast data. In this study, the performance of authors' previous models is evaluated when using weather forecast data from two online WFSs, in the following scenarios: (i) direct ET0 estimation by an ANN model, and (ii) estimate SR by ANN model, and then use that estimation for ET0 computation, using the FAO56-PM method. Employing data collected from two WFSs and a WS located in Vale do Lobo, Portugal, the latter approach achieved the best result, with a coefficient of determination (R2) ranging between 0.893 and 0.667, when considering forecasts up to 15 days., Comment: A partial version of the work submitted to ESRE/INTERNATIONAL CONFERENCE ON ENVIRONMENTAL SCIENCES AND RENEWABLE ENERGY

About 20 percent of patients undergoing breast-conserving surgery require reoperation due to cancerous tissue remaining inside the breast. Breast cancer localization systems utilize auditory feedback to convey the distance between a localization probe and a small marker (seed) implanted into the breast tumor prior to surgery. However, no information on the location of the tumor margin is provided. To reduce the reoperation rate by improving the usability and accuracy of the surgical task, we developed an auditory display using shape sonification to assist with tumor margin localization. Accuracy and usability of the interactive shape sonification were determined on models of the female breast in three user studies with both breast surgeons and non-clinical participants. The comparative studies showed a significant increase in usability (p<0.05) and localization accuracy (p<0.001) of the shape sonification over the auditory feedback currently used in surgery., Comment: 15 pages, 9 figures

The point spread function (PSF) is fundamental to any type of microscopy, most importantly so for single-molecule localization techniques, where the exact PSF shape is crucial for precise molecule localization at the nanoscale. However, optical aberrations and fixed fluorophore dipoles can lead to non-isotropic and distorted PSFs, thereby complicating and biasing conventional fitting approaches. In addition, some researchers deliberately modify the PSF by introducing specific phase shifts in order to provide improved sensitivity, e.g., for localizing molecules in 3D, or for determining the dipole orientation. For devising an experimental approach, but also for interpreting obtained data it would be helpful to have a simple visualization tool which calculates the expected PSF for the experiment in mind. To address this need, we have developed a comprehensive and accessible computer application that allows for the simulation of realistic PSFs based on the full vectorial PSF model. It incorporates a wide range of microscope and fluorophore parameters, enabling an accurate representation of various imaging conditions. Further, our app directly provides the Cramer-Rao bound for assessing the best achievable localization precision under given conditions. In addition to facilitating the simulation of PSFs of isotropic emitters, our application provides simulations of fixed dipole orientations as encountered, e.g., in cryogenic single-molecule localization microscopy applications. Moreover, it supports the incorporation of optical aberrations and phase manipulations for PSF engineering, as well as the simulation of crowded environments with overlapping molecules. Importantly, our software allows for the fitting of custom aberrations directly from experimental data, effectively bridging the gap between simulated and experimental scenarios, and enhancing experimental design and result validation.

Inspired by the notions of local equivalence in monopole and Heegaard Floer homology, we introduce a version of local equivalence that combines odd Khovanov homology with equivariant even Khovanov homology into an algebraic package called a local even-odd (LEO) triple. We get a homomorphism from the smooth concordance group $C$ to the resulting local equivalence group $C_{LEO}$ of such triples. We give several versions of the $s$-invariant that descend to $C_{LEO}$, including one that completely determines whether the image of a knot $K$ in $C_{LEO}$ is trivial. We discuss computer experiments illustrating the power of these invariants in obstructing sliceness, both statistically and for some interesting knots studied by Manolescu-Piccirillo. Along the way, we explore several variants of this local equivalence group, including one that is totally ordered., Comment: 43 pages; v2: Added mention of related work of Lewark and also Remark 5.8

We combine the recent relaxation approach with multiderivative Runge-Kutta methods to preserve conservation or dissipation of entropy functionals for ordinary and partial differential equations. Relaxation methods are minor modifications of explicit and implicit schemes, requiring only the solution of a single scalar equation per time step in addition to the baseline scheme. We demonstrate the robustness of the resulting methods for a range of test problems including the 3D compressible Euler equations. In particular, we point out improved error growth rates for certain entropy-conservative problems including nonlinear dispersive wave equations.

The GERmanium Detector Array (GERDA) experiment operated enriched high-purity germanium detectors in a liquid argon cryostat, which contains 0.33% of $^{36}$Ar, a candidate isotope for the two-neutrino double-electron capture (2$\nu$ECEC) and therefore for the neutrinoless double-electron capture (0$\nu$ECEC). If detected, this process would give evidence of lepton number violation and the Majorana nature of neutrinos. In the radiative 0$\nu$ECEC of $^{36}$Ar, a monochromatic photon is emitted with an energy of 429.88 keV, which may be detected by the GERDA germanium detectors. We searched for the $^{36}$Ar 0$\nu$ECEC with GERDA data, with a total live time of 4.34 yr (3.08 yr accumulated during GERDA Phase II and 1.26 yr during GERDA Phase I). No signal was found and a 90% C.L. lower limit on the half-life of this process was established T$_{1/2}$ > 1.5x10$^{22}$ yr, Comment: 10 pages, 5 figures, 1 table

Using mode coupling theory the conditions for all allowed dynamical universality classes for the conserved modes in one-dimensional driven systems are presented in closed form as a function of the stationary currents and their derivatives. With a view on the search for the golden ratio universality class the existence of some families of microscopic models is ruled out a priori by using an Onsager-type macroscopic current symmetry. At equal mean densities of the conserved quantities the golden modes can only appear if the currents are antisymmetric under interchange of the conserved densities and if these densities are correlated, but not in the symmetric case where at equal densities one mode is always diffusive and the second may be either Kardar-Parisi-Zhang (KPZ), modified KPZ, 3/2-L\'evy, or also diffusive. We also show that the predictions of mode coupling theory for a noisy chain of harmonic oscillators are exact., Comment: 12 pages

We show that domain walls separating coexisting extremal current phases in driven diffusive systems exhibit complex stochastic dynamics, with a subdiffusive temporal growth of position fluctuations due to long-range anticorrelated current fluctuations and a weak pinning at long times. This weak pinning manifests itself in a saturated width of the domain wall position fluctuations that increases sublinearly with the system size. As a function of time $t$ and system size $L$, the width $w(t,L)$ exhibits a scaling behavior $w(t,L)=L^{3/4}f(t/L^{9/4})$, with $f(u)$ constant for $u\gg1$ and $f(u)\sim u^{1/3}$ for $u\ll1$. An Orstein-Uhlenbeck process with long-range anticorrelated noise is shown to capture this scaling behavior. Results for the drift coefficient of the domain wall motion point to memory effects in its dynamics., Comment: 6 pages, 4 figures plus 6 pages supplemental material with 3 figures

Spectroscopic photoemission microscopy is a well-established method to investigate the electronic structure of surfaces. In modern photoemission microscopes the electron optics allows imaging of the image plane, momentum plane, or dispersive plane, depending on the lens setting. Furthermore, apertures allow filtering of energy-, real-, and momentum space. Here, we describe how a standard spectroscopic and low energy electron microscope can be equipped with an additional slit at the entrance of the already present hemispherical analyzer to enable an angle- and energy resolved photoemission mode with micrometer spatial selectivity. We apply a photogrammetric calibration to correct for image distortions of the projective system behind the analyzer and present spectra recorded on Au(111) as a benchmark. Our approach makes data acquisition in energy-momentum space more efficient, which is a necessity for laser-based pump-probe photoemission microscopy with femtosecond time resolution., Comment: This manuscript has been peer-reviewed and is currently undergoing revision

About: We propose an incremental LOD generation approach for point clouds that allows us to simultaneously load points from disk, update an octree-based level-of-detail representation, and render the intermediate results in real time while additional points are still being loaded from disk. LOD construction and rendering are both implemented in CUDA and share the GPU's processing power, but each incremental update is lightweight enough to leave enough time to maintain real-time frame rates. Background: LOD construction is typically implemented as a preprocessing step that requires users to wait before they are able to view the results in real time. This approach allows users to view intermediate results right away. Results: Our approach is able to stream points from an SSD and update the octree on the GPU at rates of up to 580 million points per second (~9.3GB/s from a PCIe 5.0 SSD) on an RTX 4090. Depending on the data set, our approach spends an average of about 1 to 2 ms to incrementally insert 1 million points into the octree, allowing us to insert several million points per frame into the LOD structure and render the intermediate results within the same frame. Discussion/Limitations: We aim to provide near-instant, real-time visualization of large data sets without preprocessing. Out-of-core processing of arbitrarily large data sets and color-filtering for higher-quality LODs are subject to future work.

Smart grid interoperability is the means to achieve the twin green and digital transition but re-mains heterogeneous and fragmented to date. This work presents the first ideas and corner-stones of an Interoperability Roadmap for the Energy Transition that is being developed by the Horizon Europe int:net project. This roadmap builds on four cornerstones that address open interoperability issues. These are a knowledge base to address the lack of convergence among existing initiatives, a maturity model and a network of testing and certification facilities to ad-dress the lack of practical tools for the industry, and a governance process to address the gap between standards-related approaches of Standards Development Organisations and Research and Innovation projects. A community of practice will be set up to ensure the continuity of the ongoing activities related to smart grid interoperability. To outlive the duration of the int:net project, the aim is to formalise the community of practice as a legal entity., Comment: 12. (Hybrid) Symposium Communications for Energy Systems (ComForEn 2023)

Data augmentation (DA) is a key factor in medical image analysis, such as in prostate cancer (PCa) detection on magnetic resonance images. State-of-the-art computer-aided diagnosis systems still rely on simplistic spatial transformations to preserve the pathological label post transformation. However, such augmentations do not substantially increase the organ as well as tumor shape variability in the training set, limiting the model's ability to generalize to unseen cases with more diverse localized soft-tissue deformations. We propose a new anatomy-informed transformation that leverages information from adjacent organs to simulate typical physiological deformations of the prostate and generates unique lesion shapes without altering their label. Due to its lightweight computational requirements, it can be easily integrated into common DA frameworks. We demonstrate the effectiveness of our augmentation on a dataset of 774 biopsy-confirmed examinations, by evaluating a state-of-the-art method for PCa detection with different augmentation settings., Comment: Accepted at MICCAI 2023

The Fibonacci sequence, $F_n = F_{n - 1} + F_{n - 2}$, and its counterpart for $n < 0$, the negaFibonacci sequence, $F_{-n} = (-1)^{n + 1} \cdot F_n$, are among the most studied sequences in mathematics. In this paper we will present a new kind of sequence, the negaFibonacci-esque sequences, identified by the property that $\omega_n = \omega_{n - 2} - \omega_{n - 1}$ with $\omega_1, \omega_2 \in \mathbb{C}$ chosen at will. We will partition this kind into natural complete and complex complete negaFibonacci-esque sequences.We will prove that $\frac{1}{\phi^n}$, referred to as the principal negaFibonacci-esque sequence, is not only a complex complete negaFibonacci-esque sequence but also one of the most significant. Furthermore, we will present an explicit formula for all complex complete negaFibonacci-esque sequences constructed with a combination of two negaFibonacci terms. We shall then conclude by connecting these sequences to the golden spiral and ratio., Comment: 9 pages, 1 figure, Fibonacci Quaterly

We propose and implement a family of quantum-informed recursive optimization (QIRO) algorithms for combinatorial optimization problems. Our approach leverages quantum resources to obtain information that is used in problem-specific classical reduction steps that recursively simplify the problem. These reduction steps address the limitations of the quantum component and ensure solution feasibility in constrained optimization problems. Additionally, we use backtracking techniques to further improve the performance of the algorithm without increasing the requirements on the quantum hardware. We demonstrate the capabilities of our approach by informing QIRO with correlations from classical simulations of shallow (depth $p=1$) circuits of the quantum approximate optimization algorithm (QAOA), solving instances of maximum independent set and maximum satisfiability problems with hundreds of variables. We also demonstrate how QIRO can be deployed on a neutral atom quantum processor available online on Amazon Braket to find large independent sets of graphs. In summary, our scheme achieves results comparable to classical heuristics, such as simulated annealing and greedy algorithms, even with relatively weak quantum resources. Furthermore, enhancing the quality of these quantum resources improves the performance of the algorithms, highlighting the potential of QIRO. Notably, the modular nature of QIRO offers various avenues for modifications, positioning our work as a blueprint for designing a broader class of hybrid quantum-classical algorithms for combinatorial optimization., Comment: 17 pages, 10 figures. Revised Figs. 1 and 2. Accepted version

We present the measurement of the two-neutrino double-$\beta$ decay rate of $^{76}$Ge performed with the GERDA Phase II experiment. With a subset of the entire GERDA exposure, 11.8 kg$\cdot$yr, the half-life of the process has been determined: $T^{2\nu}_{1/2} = (2.022 \pm 0.018_{stat} \pm 0.038_{sys})\times10^{21}$ yr. This is the most precise determination of the $^{76}$Ge two-neutrino double-$\beta$ decay half-life and one of the most precise measurements of a double-$\beta$ decay process. The relevant nuclear matrix element can be extracted: $M^{2\nu}_{\text{eff}} = (0.101\pm0.001).$, Comment: 7 pages, 4 figures, 2 tables

The fifth-order KP II equation $$ \partial_t u + \alpha \partial_x^3 u + \beta \partial_x^5 u + u \partial_x u + \partial_x^{-1} \partial_y^2u=0$$ ($\beta<0$, $\alpha>0$) is a nonlinear dispersive equation that models long dispersive waves in two space dimensions. We prove that solutions of the fifth-order KP II equation scatter to solutions of the corresponding linear equation $$ \partial_t v + \alpha\partial_x^3 v + \beta\partial_x^5 v + \partial_x^{-1} \partial_y^2 v = 0$$ for small data. Our proof uses builds on Hadac, Herr, and Koch's work (see ArXiv:0708.2011) on the third-order KP II equation., Comment: 28 pages, no figures

Here, we report a novel and conceptually straightforward technique to detect the spin-polarization directly, with laser spot spatial resolution, at buried ferromagnet/insulator interfaces in ambient settings. This has been accomplished by monitoring the voltage-induced change of the longitudinal MOKE signal as a function of the applied magnetic field and applying an AC voltage across the interface. For the case where the spin polarization enters the VMOKE signal, a simple quantitative model is proposed. A distinct positive majority spin polarization has been found for Fe and Co, whereas Ni exhibits a negative minority spin polarization.

In this work, we develop a class of high-order multiderivative time integration methods that is able to preserve certain functionals discretely. Important ingredients are the recently developed Hermite-Birkhoff-Predictor-Corrector methods and the technique of relaxation for numerical methods of ODEs. We explain the algorithm in detail and show numerical results for two- and three-derivative methods, comparing relaxed and unrelaxed methods. The numerical results demonstrate that, at the slight cost of the relaxation, an improved scheme is obtained., Comment: Submitted to the Proceedings in Applied Mathematics and Mechanic (GAMM Annual Meeting 2023)

We search for tri-nucleon decays of $^{76}$Ge in the dataset from the GERmanium Detector Array (GERDA) experiment. Decays that populate excited levels of the daughter nucleus above the threshold for particle emission lead to disintegration and are not considered. The ppp-, ppn-, and pnn-decays lead to $^{73}$Cu, $^{73}$Zn, and $^{73}$Ga nuclei, respectively. These nuclei are unstable and eventually proceed by the beta decay of $^{73}$Ga to $^{73}$Ge (stable). We search for the $^{73}$Ga decay exploiting the fact that it dominantly populates the 66.7 keV $^{73m}$Ga state with half-life of 0.5 s. The nnn-decays of $^{76}$Ge that proceed via $^{73m}$Ge are also included in our analysis. We find no signal candidate and place a limit on the sum of the decay widths of the inclusive tri-nucleon decays that corresponds to a lower lifetime limit of 1.2x10$^{26}$ yr (90% credible interval). This result improves previous limits for tri-nucleon decays by one to three orders of magnitude., Comment: 8 pages, 9 figures, 3 tables